Second-countable space

Abstract: We characterize minimal open sets in topological spaces. We show that any nonempty subset of a minimal open set is pre-open. As an application of a theory of minimal open sets, we obtain a sufficient condition for a locally finite space to be a pre- Hausdorff space. 2000 Mathematics Subject Classification. 54A05, 54D99. 1. Introduction. Let X be a topological space. We call a nonempty open set U of X a minimal open set when the only open subsets of U are U and ∅. In this paper, we study fundamental properties of minimal open sets and apply them to obtain some results on pre-open sets (cf. (2)) and pre-Hausdorff spaces. In Section 2, we characterize minimal open sets, that is, we show that a nonempty open set U is a minimal open set if and only if Cl(U) = Cl(S) for any nonempty subset S of U. This result implies that any nonempty subset S of a minimal open set U is a pre-open set. In Section 3, we study minimal open sets in locally finite spaces. The results of this section are closely related to the work of James (1), and these results will be used in the next scetion. In Section 4, we apply the theory of minimal open sets to study pre-open sets. Our first main result of this section is a property of the set of all minimal open sets in any nonempty finite open set which is not a minimal open set. This result enables us to prove a generalization of Theorem 2.5, when U is a nonempty finite open set, in Theorem 4.4. Theorem 4.5 shows that our theory of minimal open set is useful to study pre-open sets. Finally, we show that some conditions on minimal open sets implies pre-Hausdorff- ness of a space, that is, if any minimal open set of a locally finite space X has two elements at least, then X is a pre-Hausdorff space.

Abstract: If (G, 8) is a second countable transformation group and the stability groups are amenable then C*(G, 8) is C.C.R. if and only if the orbits are closed and the stability groups are C.C.R. In addition, partial results relating closed orbits to C.C.R. algebras are obtained in the nonseparable case. In several cases, the topology of the primitive ideal space is calculated explicitly. In particular, if the stability groups are all contained in a fixed abelian subgroup H, then the topology is computed in terms of H and the orbit structure, provided C*(G, 8) and C*(H, 8) are EH-regular. These conditions are automatically met if G is abelian and (G, 8) is second countable.